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TRIGONOMETRIC FUNCTIONS.

DIRECTIONS FOR USING THE TABLE. The table given on pages 74-78 contains the natural sines, cosines, tangents, and cotangents of angles from 0° to 90°. Angles less than 45o are given in the first column at the lefthand side of the page, and the names of the functions are given at the top of the page; angles greater than 45o appear at the right-hand side of the page, and the names of the functions are given at the bottom. Thus, the second column contains the sines of angles less than 45o and the cosines of angles greater than 45o; the sixth column contains the cotangents of angles less than 45o and the tangents of angles greater than 45°. To find the function of an angle less than 45°, look in the column of angles at the left of the page for the angle, and at the top of the page for the name of the function; to find a function of an angle greater than 45°, look in the column at the right of the page for the angle and at the bottom of the page for the name of the function. The successive angles differ by an interval of 10'; they increase downwards in the left-hand column and upwards in the right-hand column. Thus, for angles less than 45o read down from top of page, and for angles greater than 45o read up from bottom of page.

The third, fifth, seventh, and ninth columns, headed d, contain the differences between the successive functions; for example, in the second column we find that the sine of 320 10 is .5324 and that the sine of 32° 20' is .5348; the difference is .5348-.5324 .0024, and the 24 is written in the third column, just opposite the space between .5324 and .5348. In like manner the differences between the successive tabular values of the tangents are given in the fifth column, those between the cotangents in the seventh column, and those for the cosinės in the ninth column. These differences in the functions correspond to a difference of 10 in the angle; thus, when the angle 32° 10' is increased by 10, that is, to 32° 20', the increase of the sine is .0024, or, as given in the table, 24. It will be observed that in the tabular difference no attention is paid to the decimal point, it being understood that the difference is merely the number obtained by subtracting the last two or three figures of the smaller function from those of the larger. These differences are used to obtain the sines, cosines, etc. of angles not given in the table; the method employed may be illustrated by an example. Required, the tangent of 27° 34'. Looking in the table, we see that the tangent of 270 30 is .5206, and (in column 5) the difference for 10' is 37. Difference for 1' is 37 • 10 = 3.7, and difference for 4' is 3.7 X 4 = 14.8. Adding this difference to the value of the tan 27° 30', we have

tan 27° 30 .5206 difference for 4! 14.8

tan 27° 31' = .5220.8 or .5221, to four places. Since only four decimal places are retained, the 8 in the fifth place is dropped and the figure in the fourth place is increased by 1, because 8 is greater than 5.

To avoid multiplication, the column of proportional parts, headed P. P., at the extreme right of the page, is used. At the head of each table in this column is the difference for 10, and below are the differences for any intermediate number of minutes from 1' to 9'. In the above example, the difference for 10' was 37; looking in the table with 37 at the head, the difference opposite 4 is 14.8; that opposite 7 is 25.9; and so on. For want of space, the differences for the cotangents for angles less than 45° (or the tangents of angles greater than 45°) have been omitted from the tables of proportional parts. The use of these functions should be avoided, if possible, since the differences change very rapidly, and the computation is therefore likely to be inexact. The method to be employed when dealing with these functions may be shown by an example: Required, the tangent of 76° 34'. Since this angle is greater than 45°, we look for it in the column at the right, and read up; opposite the 76° 30', we find, in sixth column, the number 4.1653, and corresponding to it in seventh column is the difference 540. Since 540 is the difference for 10, the difference for 4' is 540 X 14 216. Adding this difference:

tan 760 30 4.1653 difference for 4! 216

tan 76° 31' = 4.1869

When the angle contains a certain number of seconds, divide the number by 6, and take the whole number nearest to the quotient; look out this number in the table of proportional parts (under the proper difference), and take out the number that is opposite to it. Shift the decimal point one place to the left, and then add it to the partial function already found. Find the sine of 34° 26' 44". sine 34° 20 =.5640

Difference for 10% = 24. difference for 6' 14.4 difference for 44" 1.7 4 = 71. Look out in the P. P.

table the number under 24 sine 34° 26 44" =.5656

and opposite 7.

It is 16.8. Shifting the decimal point one place to the left, we get

1.68, or, say, 1.7. The tangent is found in the same way as the sine. To find the cosine of an angle:

As the angle increases, the value of the cosine decreases, 80 that, instead of adding the values corresponding to 6' and 44" to the function already found, we subtract them from it. Thus, find cos 34° 26' 44".

cos 34° 20 = .8258 Difference for 10 = 17.

difference for 6' 10.2 difference for 44" 1.2 The number under the 17 and

opposite the 7, in the P. P. total difference 11.4

table, is 11.9. Therefore, take .8247 1.19, or, say,

1.2. Therefore, cos 34° 26' 44" .8258 - .0011 = .824°.

Only four decimal places are kept; therefore, the figure of the difference following the decimal point is dropped before subtracting

The cotangent is found in the same manner.
We will now consider angles greater than 45o.
Find the sine of 68° 47' 22'.

In obtaining the difference, it must be remembered to choose the one between the sine of 68° 40' and the next angle above it, namely, 680 50'.

1.1

sine 68° 40 .9315 Difference for 10 = 10. difference for 7

7 difference for 191 = .4 4 = 3f, say 4. Under the 10 sine 68° 47' 2211 = .9322

and opposite the 4 is the
number 4.0; shifting the deci.

mal point, we get .4.
As usual, only four decimal places are kept.
The tangent is found in the same manner.
Find cos 68° 47' 22".

As before, the cosine decreases as the angle increases; therefore, we subtract the successive sine values corresponding to the increments in the angle.

cos 68° 40 = .3638 Difference for 10 = 27. difference for 7' 18.9 difference for 22''

Under the 27 and opposite the total difference 20

4 is the number 10.8; there

fore, take 1.08 in this case, .3618

or, say, 1.1. Therefore, cos 68° 47' 22'' .3638 - .002 = .3618. The cotangent is found in the same way.

In finding the functions of an angle, the only difficulty likely to be encountered is to determine whether the difference obtained from the table of proportional parts is to be added or subtracted. This can be told in every case by observing whether the function is increasing or decreasing as the angle increases. For example, take the angle 21°; its sine is .3584, and the following sines, reading downwards, are .3611, .3638, etc. It is plain, therefore, that the sine of say 21° 6' is greater than that of 21°, and that the difference for 6' must be added. On the other hand, the cosine of 21° is .9336, and the following cosines, reading downwards, are .9325, .9315, etc.; that is, as the angle grows larger the cosine decreases. The cosine of an angle between 21° and 21° 10', say 21° 6', must therefore lie between .9325 and .9315; that is, it must be smaller than .9325, which shows that in this case the difference for 6' must be subtracted from the cosine of 210.

We will now consider the case in which the function, i.e., the sine, cosine, tangent, or cotangent, is given and the corresponding angle is to be found.

Find the angle whose sine is .4943. The operation is Arranged as follows:

.4943

Difference for 10 = 26. .4924 = sin 29° 30.

1st remainder

19
18.2

= difference for 7'.

.78 =

2d remainder .8

difference for .3 or 18". .4943 sin 29° 37' 18".

Looking down the second column, we find the sine next smaller than .4943 to be .4924, and the difference for 10 to be 26. The angle corresponding to .4924 is 290 30'. Subtracting the .4924 from .4943, the first remainder is 19; looking in the table of proportional parts, the part next lower than this difference is 18.2, opposite which is 7'. Subtracting this difference from the remainder, we get .8, and, looking in the table, we see that 7.8 with its decimal point moved one place to the left is nearest to the second difference. This is the difference for .3' or 18'. Hence, the angle is 29° 30' +7

290 37' 18''. Find the angle whose tangent is .8824.

.8824

Difference for 10 = 51. .8796 = tan 41° 20'.

+ 18

1st remainder , 28

25.5

difference for 5'.

2d remainder 2.5

2.55 difference for .5' or 30". .8824 = tan 41° 25' 30''.

In the two examples just given, the minutes and seconds corresponding to the 1st and 2d remainders are added to the angle taken from the table. Thus, in the first example, an inspection of the table shows that the angle increases as the sine increases; hence, the angle whose sine is .4943 must be greater than 290 30', whose sine is .4924. For this reason the correction must be added to 290 30'. The same reasoning applies to the second example.

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